Galois Theory Challenges Weisfeiler Leman: Invariant Features for Symmetric Matrices and Point Clouds

Beatrix Yaxin Wen, Caio Netto, Thabo Samakhoana, Soledad Villar, Ningyuan Huang

Keywords: Expressivity, Galois theory, Weisfeiler Leman, graph neural networks

Abstract: Recent work Blum-Smith et al. [2024] proposed an invariant machine learning model on point clouds and symmetric matrices based on invariant features. For symmetric matrices, the model is invariant under the group action of permutation by conjugation. Therefore, this model can be used to test whether two graphs are isomorphic (by checking whether the invariant features of two graphs coincide). In this work, we investigate the expressive power of this new method. Our theoretical results show that on undirected graphs, the method is strictly worse than graph neural networks (GNNs) based on message passing. To improve its expressivity, we propose a modified method by adding a new invariant function, which we test empirically against GNNs and other baseline methods. The newly proposed function set outperformed the original invariant features, yielding compatible or exceeding results compared to GNNs.

Submission Number: 104